Systems and methods for decentralized attribution of generative models

ABSTRACT

A system and associated methods for decentralized attribution of GAN models is disclosed. Given a group of models derived from the same dataset and published by different users, attributability is achieved when a public verification service associated with each model (a linear classifier) returns positive only for outputs of that model. Each model is parameterized by keys distributed by a registry. The keys are computed from first-order sufficient conditions for decentralized attribution. The keys are orthogonal or opposite to each other and belong to a subspace dependent on the data distribution and the architecture of the generative model.

CROSS REFERENCE TO RELATED APPLICATIONS

This is a U.S. Non-Provisional patent application that claims benefit to U.S. provisional patent application Ser. No. 63/122,306 filed on Dec. 7, 2020, which is herein incorporated by reference in its entirety.

FIELD

The present disclosure generally relates to artificial intelligence and associated issues thereof; and in particular, to decentralization attribution of generative models.

BACKGROUND

There have been growing concerns regarding the fabrication of content through generative models. Specifically, for example, recent advances in generative models have enabled the creation of synthetic content that are indistinguishable even by naked eyes. Such successes raised serious concerns regarding adversarial applications of generative models, e.g., for the fabrication of user profiles, articles, images, audios, and videos. Necessary measures have been called for the filtering, analysis, tracking, and prevention of malicious applications of generative models before they create catastrophic sociotechnical damages. In particular, a need exists for attribution of machine-generated contents back to its source model to facilitate IP protection and content regulation.

It is with these observations in mind, among others, that various aspects of the present disclosure were conceived and developed.

SUMMARY

Aspects of the present disclosure may take the form of a system for decentralized attribution of generative models, and/or methods thereof. In some examples, the system includes a processor configured with instructions to provide a registry and verification service that improves attribution of a generative adversarial network (GAN) relative to other versions of the GAN. Specifically, the processor accesses a dataset and a GAN associated with the dataset, computes a plurality of keys and a plurality of corresponding GANs such that at least one key is computed for each GAN of the plurality of GANs, each GAN of the plurality of GANs being a version of the GAN modified by the at least one key, wherein the plurality of keys are derived from first-order sufficient conditions for decentralized attribution, and the processor verifies a GAN of the plurality of GANs based upon an output of a query associated with the GAN.

The present disclosure may further take the form of a tangible, non-transitory, computer-readable media having instructions encoded thereon, the instructions, when executed by a processor, being operable to: compute a sequence of keys by a registry for key-dependent GAN models, the keys configured for strict data compliance and orthogonality so as to accommodate tracing of machine-generated contents back to its source model, wherein the keys are orthogonal or opposite to each other and belong to a subspace dependent on the data distribution and the architecture of the generative model.

Other examples are contemplated and supported by the disclosure described herein.

BRIEF DESCRIPTION OF THE DRAWINGS

The patent or application file contains at least one drawing executed in color. Copies of this patent or patent application publication with color drawing(s) will be provided by the Office upon request and payment of the necessary fee.

FIGS. 1A-1C are a series of illustrations respectively showing: (a) a protocol of decentralized attribution, keys being distributed by the registry and used to produce key-dependent generators for individual users; (b) orthogonal keys ϕ₁ and ϕ₂) achieve distinguishability and attributability; and (c) acute keys achieve distinguishability but not attributability.

FIGS. 2A and 2B are respective graphical illustrations of: (a) Eigenvectors for the two largest and two smallest eigenvalues of M for DCGANS on MNIST (top) and CelebA (bottom); and (b) Left to Right: Samples from G_(D) and subtraction of G_(D) G_(eigenvalues).

FIGS. 3A and 3B are respective graphical illustrations showing (a) a graph showing d_(max) is bounded by ∥Δ_(x)∥; and o_(i) are close to 0; and (b) an illustration showing Eigenvectors and the corresponding samples from (top to bottom) the largest eigenvector of third layer and last layer of M_(MNIST), the largest eigenvector of third layer and last layer of M_(CelebA).

FIGS. 4A-4F are a series of images respectively showing: (a) 1st-2rd row: samples from G_(D) and non-robust generator (b-f) 1st-2rd rows: worst case post-process, samples from robust training against the specific post-processes (prior to the post processes). 3^(rd) row for all: differences between 2^(nd) row of (a) and 2^(nd) row of each image. As a result, we can reveal the changes in the attributions.

FIGS. 5A and 5B are Scree plots. Most of the eigenvalues are close to 0.

FIGS. 6A and 6B are respective graphical representation showing: (a) d_(max) is bounded by ∥Δ_(x)∥ and o_(i) is close to 0; and (b) o_(i) and FID show positive correlation (0.63). Also, D(G_(ϕ)

and A({G_(ϕ) _(i) }_(x−1) ^(i)

are close to 1.

FIGS. 7A and 7B are graphical representations showing: (a) Largest eigenvectors of the first layer to last layer (top to bottom) and corresponding samples (cosine similarities with largest eigenvector of M are −0.49, 0.20, −0.98, 0.49); and (b) Largest eigenvectors of the first layer to the last layer (top to bottom) and corresponding samples (cosine similarities with largest eigenvector of M are 0, 0.01, −0.05, and −0.5).

FIGS. 8A-8F are a series of images showing DCGAN-MNIST.

FIGS. 9A-9F are a series of images showing DCGAN-CelebA.

FIGS. 10A-10F are a series of images showing CycleGAN-Cityscapes.

FIG. 11 is a simplified block diagram of a computer-implemented system associated with the decentralized attribution of generative models concepts described herein.

FIG. 12 is a simplified block diagram of an exemplary process associated with the decentralized attribution of generative models concepts described herein.

FIG. 13 is an exemplary simplified block diagram of a computing device that may be configured to implement various methodologies described herein.

Corresponding reference characters indicate corresponding elements among the view of the drawings. The headings used in the figures do not limit the scope of the claims.

DETAILED DESCRIPTION

There have been growing concerns regarding the fabrication of contents such as realistic-appearing photos and human faces through generative models. The present disclosure investigates the feasibility of decentralized attribution of fabricated content to such models. Given a group of models derived from the same dataset and published by different users, attributability of a generative model is achieved when a public verification service associated with each model (a linear classifier) returns positive only for outputs of that model. Attribution allows tracing of machine-generated contents back to its source model, thus facilitating IP-protection and content regulation. Decentralized attribution prevents forgery of source models by only allowing users to have access to their own classifiers, which are parameterized by keys distributed by a registry. One notable feature of the present disclosure is the development of design rules for the keys, which are derived from first-order sufficient conditions for decentralized attribution. Through validation on MNIST, CelebA and Cityscapes, it is shown that keys may be (1) orthogonal or opposite to each other and (2) belonging to a subspace dependent on the data distribution and the architecture of the generative model. This paper also empirically examines the trade-off between generation quality and robust attributability against adversarial post-processes of model outputs.

INTRODUCTION

Existing studies primarily focused on the detection of machine-generated contents. Marra et al., incorporated by reference in its entirety, (“Do gans leave artificial fingerprints?” In 2019 IEEE Conference on Multimedia Information Processing and Retrieval (MIPR), pages 506-511. IEEE, 2019) showed empirical evidence that generative adversarial networks (GANs) may come with data-specific fingerprints in the form of averaged residual over the generated distribution, yet suggested that generative models trained on similar datasets may not be uniquely distinguishable through fingerprints. Yu et al., incorporated by reference in its entirety, (“Attributing fake images to gans: Analyzing fingerprints in generated images.” arXiv preprint arXiv:1811.08180, 2018) showed on the other hand that it is empirically feasible to attribute a finite and fixed set of GAN models derived from the same dataset, i.e., correctly classifying model outputs by their associated GANs. While encouraging, their study did not prove that attribution can be achieved when the model set continues to grow (e.g., when GAN models are distributed to end users in the form of mobile apps). In fact, Wang et al., incorporated by reference in its entirety, (“Cnn-generated images are surprisingly easy to spot . . . for now.” arXiv preprint arXiv: 1912.11035, 2019) showed that detectors trained on one generative model are transferable to other models trained on the same dataset, indicating that individually trained detectors may perform incorrect attribution, e.g., by attributing images from one model belonging to user A to another model belonging to user B. It should be highlighted that most of the existing detection mechanisms are centralized, i.e., the detection relies on a registry that collects all models and/or model outputs and empirically look for collection-wise features that facilitate detection. This fundamentally limits the scalability of detection tools in real-world scenarios where an ever growing number of models are being developed even for the same dataset.

Problem Formulation

A motivation intuitively exists to investigate the feasibility of a decentralized approach to ensuring the correct attribution of generative models. Specifically, one can assume that for a given dataset

, the registry only distributes keys, Φ: −{ϕ₁, ϕ₂, . . . }, to users of generative models without collecting information from the users' models. Each key is held privately by a user, whose key-dependent model is denoted by G_(ϕ)(•:0):

^(d) ^(z) >

^(d) ^(x) where

and x are latent and output variables, respectively, and d_(z) and d_(x) the corresponding dimensionalities. θ are the model parameters. When necessary, θ and ϕ are suppressed to reduce notational burden. The distribution of each key is accompanied by that of a public verification service, which tells whether a query belongs to G_(ϕ) (labeled as 1) or not (labeled as −1). The underlying binary classifier is called as a verifier and denoted by f_(ϕ):

^(d) ^(x) >{1,1}. In some embodiments of the present disclosure, the focus is on linear classifiers: f_(ϕ)(x)=sign(ϕ^(T)x). In one example: Consider that a registry (run by a company) develops a new GAN model for photo post-processing. Individuals download the app that includes a GAN model and a key. The installation modifies the GAN according to the keys so that the resulting model can be verified. The keys are then deleted from the users' end. All outputs from the user-end models can now be traced back to the users (FIG. 1).

The following quantities are central to investigation: The distinguishability of G_(ϕ) is defined as

$\begin{matrix} {{{D\left( G_{o} \right)}:{- {E_{{{.r} \sim P_{C}},{{\ldots\mspace{11mu} r^{\prime}} \sim P_{D}}}\begin{bmatrix} {\begin{matrix} 1 \\ 2 \end{matrix}\left( {{f_{\phi}\left( {.r} \right)} - 1} \right)} & \vdash & {\begin{matrix} 1 \\ 2 \end{matrix}\left( {{f_{\phi}\left( {.r^{\prime}} \right)} - 1} \right)} \end{bmatrix}}}},} & (1) \end{matrix}$

where

is the authentic data distribution, and P_(G) _(ϕ) the model distribution induced by G_(ϕ). The attributability of a collection of generative models

:={G₁ . . . G_(N)} is defined as:

$\begin{matrix} {{A{()}}:={\sum\limits_{i = 1}^{N}{\frac{1}{2N}\left( {{\text{?}\left\lbrack {{{\mathbb{I}}\left( {{\text{?}\left( \text{?} \right)} = 1} \right\rbrack} + {\frac{1}{N - 1}\text{?}{\text{?}\left\lbrack {1\left( {{\text{?}\left( \text{?} \right)} = {- 1}} \right)} \right\rbrack}}} \right)}\text{?}\text{indicates text missing or illegible when filed}}\mspace{245mu} \right.}}} & (2) \end{matrix}$

Distinguishability of G (attributability of

) is achieved when D(G)=1 (A(

)=1). Lastly, a root model sent to all users along the key is denoted by G(•:θ₀) (or shortened as G₀), and it is further assumed that P_(G) ₀ −

. A (lack of) generation quality of G_(ϕ) is measured through both the Fréchet Inception Distance (FID) score and the l₂ norm of the mean output perturbation

Δx(ϕ)=

_(z˜P) _(z) [G _(ϕ)(z:θ)−G(z:θ ₀)]  (3)

where P_(z) is the latent distribution.

The present disclosure answers the following question: What are the rules for designing keys, so that the resultant generative models can achieve distinguishability individually and attributability collectively?

Contributions—Features and Advantages

It is believed the present disclosure provides the following contributions:

First-order sufficient conditions for distinguishability and attributability are developed to connect the aforementioned metrics with a geometry of the data distribution, a sensitivity of the generative model, angles between keys, and the generation quality.

The sufficient conditions yield simple design rules for the keys, which should be (1) data compliant, i.e., f_(ϕ)(x)=−1 for x˜

, (2) orthogonal or opposite to each other, and (3) within a model- and data-dependent subspace to maintain generation quality.

This disclosure empirically validates the design rules and studies the capacity of keys using Deep Convolutional GAN (DCGAN), Probabilistic GAN (PGAN), and CycleGAN on MNIST (database), CelebA, and the Cityscape datasets.

Additionally, this disclosure empirically tests tradeoffs between generation quality and robust attributability under post-processes including image blurring, cropping, noising, JPEG conversion, and a combination of all, and shows that robust attributability can be achieved, with degraded yet acceptable generation quality.

Notations. Throughout the present disclosure, the i^(th) element of vector a is denoted by a_((i)), and A_((i,j)) the (i, j)th element of matrix A. ∥a∥_(H) ²−a^(T)Ha for vector a and matrix H. ∇_(x)y|_(x) _(o) is the gradient of

with respect to x, evaluated at x=x₀ Supp(P) is used to denote the support of distribution P.

Key Design for Distinguishability, Attributability, and Generation Quality

A Toy Case

Connections among distinguishability, attributability, and generation quality are illustrated through a toy case with the following settings: (1) One-hot orthogonal keys: Let ϕi_(i)⊂Φ be one-hot and ϕ^(T)ϕ′−0 for all ϕ≠ϕ′. (2) Data compliance: Let x˜

have negative elements so that f_(ϕ)(x)−1 for all x, i.e., the authentic data is correctly attributed by all verifiers as not belonging to their associated generators. (3) Distinguishability through output perturbation: A key-dependent generative model G_(ϕ) achieves distinguishable output distribution P_(G) _(ϕ) by adding a fixed and bounded perturbation δ to the output of the root model G_(ϕ):

$\begin{matrix} {{{\min\limits_{{\delta } \leq \epsilon}{\text{?}\left\lbrack {\max\left\{ {1 - {\left( {+ \delta} \right)^{T}0}} \right\}} \right\rbrack}},{\text{?}\text{indicates text missing or illegible when filed}}}\mspace{295mu}} & (4) \end{matrix}$

where ε>0. The solution to Eq. (4) is δ*(ϕ)=ε sign(ϕ)=εϕ, which yields ∥Δx∥−∥δ*∥−ε. With these settings, we have the following proposition (proof provided below):

Proposition 1. (Toy case) If

∥Δx∥>max_(x˜)

{∥x∥ _(∝) }, D(G _(ϕ))−1∀ϕ⊂Φ and A(

)−1.

While simplistic, Proposition 1 reveals that (1) the lower bound on the degradation of generation quality to suffice distinguishability is dependent on the data geometry, and (2) orthogonality of the keys ensures attributability. These properties are preserved for a more realistic case discussed below.

A More Realistic Case

A few modifications are made to the toy case: (1) Normalized keys: The system considers data-compliant keys ϕ∈

^(d) ^(x) in a convex cone, and constrain loll |ϕ∥=1 for identifiability. (2) Distinguishability through model parameter perturbation: The output perturbation in the toy case can be reverse engineered and removed when generative models are white-box to end users. Therefore, to the system perturbs model parameters instead through the following problem:

$\begin{matrix} {{\text{?}{{\text{?}\left\lbrack {\max\left\{ {1 - {\phi^{T}{G_{o}\left( {z:\theta} \right)}{.0}}} \right\}} \right\rbrack}.\text{?}}\text{indicates text missing or illegible when filed}}\mspace{295mu}} & (5) \end{matrix}$

Distinguishability. Start by a first-order analysis, where it is assumed that for a small ε, Eq. (5) is solved by a gradient descent step:

Δ? − ?𝔼_(?)[∇_(θ), r^(T)]_(θ_(o))ϕ ?indicates text missing or illegible when filed                    

with γ>0, and a linear approximation can capture the perturbation from x₀=G(z:θ0) to x=G(z:θ) for latent z=x=x₀+∇_(θ)x₀|_(θ) _(o) Δθ. Here the data-compliance condition: 1−ϕ^(T)x>0 for x˜P_(G) _(ϕ) is used for the approximation of Δθ. To reduce notational burden, denote by J(x):=∇_(θ)x^(T)|_(θ) ₀ the Jacobian of G₀ with respect to its parameters, and let

M = E_(?)[J(x)]𝔼_(?)[J(x)]^(T) ∈ ℝ^(?).?indicates text missing or illegible when filed                   

The following conjectures about J(x) and M are empirically tested:

Conjecture 1. Let the (i, j)th element of

Σ(x) = J(x)𝔼_(?)[J(x)]^(T) − M be Σ_((i, j)) ?indicates text missing or illegible when filed                    

with variance σ_(ij) ². Then Σ_((i,j)) is approximately drawn independently from

(0, σ_(ij) ²).

Conjecture 2. Denote by Λ={λ₁ . . . λ_(d) _(x) } the eigenvalues of M. For existing deep generative models, there exists a large subset of similarly small eigenvalues.

Remarks. {σ_(ij) ²} reflects the difficulty of controlling generative models: Let J_(i)(x)^(T) be the ith row of J(x) and

J ? = ? ⁡ [ J i ⁡ ( x ) ] . ⁢ ? ⁢ indicates text missing or illegible when filed ⁢

J_(i)(x) represents the sensitivity of the ith element of x˜P_(G) ₀ with respect to θ₀. Let ΔJ_(i)=J_(i)−J _(i), then

H i = ? ⁡ [ Δ ⁢ ⁢ J i ⁢ Δ i T ] ?indicates text missing or illegible when filed                    

is the variance-covariance matrix of J_(i)(x). Let Δ_(i)(x)−J_(i) ^(T)(x)Δθ be the perturbation along the ith element of x due to Δθ, and Δ_(i)=J_(i) ^(T)Δθ the expected perturbation. Lastly, let Var(Δ_(i))=∥Δθ∥_(H) ², be the variance of the perturbation. For Δθ with unit norm, we can show that Var(Δ_(i))=σ_(ij) ²/|J _(j)∥² when Δθ is chosen to maximize Δj(Δθ−J _(j)/∥J _(j)∥). Therefore, σ_(ij) ² reflects the difficulty of controlling supp(P_(G) _(o) ) through Δθ. {σ_(ij) ²} concentrates at zero for DCGANs on MNIST and CelebA.

The first-order sufficient conditions for model distinguishability is as follows (proof below):

Theorem 1. (Realistic case) Let

d_(max)(ϕ) = max_(x ∼ P_(D))ϕ^(T)x, σ²(ϕ) = Σ_(i, j)σ_(ij)²ϕ_({i})²ϕ_({j})²,

and δ_(d) be a positive number greater than

$\exp\left( {{- \frac{1}{2}}\left( \frac{\text{?}\text{?}}{\sigma(o)} \right)^{2}} \right)$ ?indicates text missing or illegible when filed                    

for a data-compliant key ϕ∈Φ. If

$\begin{matrix} {{{{\Delta\text{?}(\phi)}} \geq {{\begin{matrix} {{d_{\max}\left( \text{?} \right)}\sqrt{\text{?}\text{?}}} \\ {{\text{?}\text{?}} - {\sigma\sqrt{\log\left( \frac{1}{\text{?}} \right)}}} \end{matrix}.\text{?}}\text{indicates text missing or illegible when filed}}}\mspace{256mu}} & (6) \end{matrix}$

then D(G_(ϕ))≥1−δ_(d)/2

Remarks. Theorem 1 reveals the connection between distinguishability and generation quality: In addition to the data geometry (d_(max)) as in the toy case, the lower bound of the generation quality also depends on model-related properties (M and σ). It should be noted that the lower bound is over approximated when a ϕ^(T) Mϕ is small: Specifically, it is shown below empirically that distinguishability can be achieved even when ϕ^(T) Mϕ, is small. We hypothesize that this is due to the nonlinear change of σ(ϕ) along the gradient descent process.

Generation quality. Note that the mean perturbation following the first-order analysis is

Δ x = 𝔼_(x₀ ∼ P_(G_(U)))[x − x₀] = 𝔼_(x ∼ P_(G_(U)))[γ J(x)𝔼_(x ∼ P_(G_(U)))[J(x)]ϕ] = γ Mϕ.

We verify through experiments that for ϕ that are eigenvectors of M, Δx×ϕ (FIG. 2B). These together with Theorem 1 lead to the following conjecture consistent with intuition, again tested through experiments (shown below):

Conjecture 3. ∥Δx∥<Σd_(max), where τ is finite and dependent on the condition number of M.

There are two aspects of generation quality that we care about: First, for ∥Δx∥ to be small, Conjecture 3 suggests that we should pick ϕ with small d_(max). Second, Spectral analysis of M for MNIST and CelebA shows that ϕs corresponding to large eigenvalues have more structured patterns, while those for small eigenvalues resemble white noise. As a result, keys in the eigenspace of small eigenvalues of M achieve better FID scores and are preferred for maintaining the salient contents of the authentic data. FIG. 2A shows the eigenvectors of the largest and smallest eigenvalues of M for DCGANs on MNIST and CelebA. FIG. 2B are the outputs of the corresponding models that achieve distinguishability.

Attributability. The first-order sufficient conditions for attributability are as follows (proof below):

Theorem 2. Let

$\begin{matrix} {{{{d_{\min}^{*} - \text{?}}❘{{{{\text{?}{\text{?}.{\overset{\_}{\sigma}}^{2}}} - {\sqrt{\text{?}V^{T}\text{?}\left( {\text{?}\text{?}} \right)^{2}}\mspace{14mu}{where}\mspace{14mu} V_{({i,j})}} - {{\sigma_{ij}^{2}.{\mspace{11mu}\;}{If}}\mspace{14mu}{D(G)}}} \geq {1\mspace{14mu}\delta_{d}\mspace{14mu}{for}\mspace{14mu}{all}\mspace{14mu}\text{?}}} \Subset \mathcal{G}}},{{{\text{?}\text{?}} \leq {0\mspace{14mu}{for}\mspace{14mu}{all}\mspace{14mu}{\text{?}.\text{?}}}} \Subset \Phi},{{{and}\mspace{14mu}{{\Delta\text{?}\left( \text{?} \right)}}} \leq {{\frac{d_{\min}^{*}\sqrt{\text{?}M^{2}\text{?}}}{\sqrt{{\text{?}M^{2}\text{?}} - \left( {\text{?}M\text{?}} \right)^{2}} + {{\sigma\left( \text{?} \right)}\sqrt{\log\left( \frac{1}{\text{?}} \right)}}}.\text{?}}\text{indicates text missing or illegible when filed}}}}\mspace{304mu}} & (7) \end{matrix}$

for all ϕ∈Φ, then A(

)≥1−(δ_(d)+δ_(u))/2.)

Remarks. (1) Conflict exists between distinguishability and attributability: The degradation of generation quality is lower bounded for distinguishability yet upper bounded for attributability. This is because the former requires model distributions to be away from

, while the latter requires G_(ϕ) to stay away from the half spaces {x∈

^(d) ^(x) |ϕ′^(T)x>0} of all other keys ϕ′≠ϕ (see FIG. 1B).

(2) Attributability is inherently limited by the model architecture: There are two reasons for G_(ϕ) to enter {x⊂

^(d) ^(x) |ϕ′^(T)x>0} by moving away from

: (i) P_(G) _(ϕ) diverges as we perturb θ due to non-zero σ ²(ϕ); (ii) the center of support(P_(G) _(ϕ) ) moves along Mϕ rather than ϕ. In the special case where σ ²(ϕ)=0 and Mϕ∝ϕ (when M has a condition number of 1), the upper bound on ∥Δx| becomes +∞.

(3) Keys need to be strictly data compliance: When d_(min)*−0, support(

) is tangent to one of the keys. Attributability cannot be achieved unless σ ²(ϕ) and Mϕ∝ϕ.

(4) ϕ^(T)ϕ′ implies orthogonal and opposite keys: ϕ^(T)ϕ′<0 requires ϕ and ϕ′ to have an orthogonal or obtuse angle. Note that for a given vector space, the capacity of keys to satisfy ϕ^(T)ϕ′<0 for all ϕ≠ϕ′ is achieved when all keys are orthogonal or opposite to each other. Therefore, we can focus on computing orthogonal keys (and flipping their signs to get the other half).

Implementation

The above analysis suggests the following rules for designing keys: (R1) strict data compliance, (R2) orthogonality, (R3) small d_(max), and (R4) belonging to the eigenspace of M associated with small eigenvalues.

Key generation. The registry computes a sequence of keys to satisfy (R1) and (R2) for decentralized attribution:

$\begin{matrix} {{\text{?} - {\arg\text{?}{\text{?}\left\lbrack {\max\left\{ {1 - {\text{?}\left( \text{?} \right){.0}}} \right\}} \right\rbrack}} + {\text{?}{{{\text{?}\text{?}}}.\text{?}}\text{indicates text missing or illegible when filed}}}\mspace{295mu}} & (8) \end{matrix}$

The orthogonality penalty is omitted for the first key. Some remarks: (1) For fast computation of keys, we convexify Eq. (8) by removing the unit norm constraint. Each key is normalized right after solving the relaxed problem. (2)

and P_(G) _(ϕ) do not perfectly match in practice, and therefore expectations are taken over both distributions. (3) We use a hinge loss to promote strict data compliance. (4) Computation of d_(max) requires minimax, and M is not always available for deep generative models due to their large parameter space. Therefore, we do not explicitly enforce (R3) or (R4), but will use them for generation quality control (see Sec. 4).

Generative models. To train key-dependent models, Eq. (5) is relaxed by introducing a penalty on the generation quality:

$\begin{matrix} {{\text{?}{{\text{?}\left\lbrack {{\max\left\{ {{1 - \text{?}},{\left( {G,\left( {z:\theta_{i}} \right)} \right){.0}}} \right\}} + {C{{{{G_{0}(z)} - {G_{i}\left( {z:\theta_{i}} \right)}}}^{2}/\text{?}}}} \right\rbrack}.\text{?}}\text{indicates text missing or illegible when filed}}\mspace{265mu}} & (9) \end{matrix}$

The hyperparameter C is tuned through a parametric study (see Appendices K).

Robust training. Lastly, we consider the scenario where outputs are post-processed before being verified. We train a robust version of the generative models against a distribution of post-processes T:

^(d) ^(x) >

^(d) ^(x) ˜P_(T) through

$\begin{matrix} {{\text{?}{{\text{?}\left\lbrack {{\max\left\{ {1 - {\text{?}\left( {T\left( {G_{i}\left( {z:\theta_{i}} \right)} \right)} \right){.0}}} \right\}} + {C{{{{G_{0}(z)} - {G_{i}\left( {z:\theta_{i}} \right)}}}^{2}/\text{?}}}} \right\rbrack}.\text{?}}\text{indicates text missing or illegible when filed}}\mspace{304mu}} & (10) \end{matrix}$

TABLE 1 Empirical averaged distinguishability (D), attributability (A( 

 ). Δ.r and FID scores from 20 generative models for each dataset. Standard deviations reported when applicable, or omitted if ≤0.05. FID of ^(G) ⁰ (FID₀) is the baseline. FID is not applicable to CycleGAN. GANs Angle Dataset D Λ( 

 ) ∥Δ.r∥ FID₀ FID DCGAN Orthogonal MNIST 0.99 0.99 5.20(0.31) 4.98(0.15) 5.68(0.23) DCGAN 45 degree MNIST 0.99 0.64 5.63(0.39) — 5.85(0.32) DCGAN Orthogonal CelebA 0.99 0.99 4.19(0.18) 33.95(013) 52.09(2.20) DCGAN 45 degree CelebA 0,99 0.59 4.75(0.20) — 59.57(2.56) PGAN Orthogonal CelebA 0.99 0.99 9.29(0.95) 13.31(0.07) 21.62(1.73) PGAN 45 degree CelebA 0.99 0.71 12.03(1.56) — 28.84(3.37) CycleGAN Orthogonal Cityscapes 0.99 0.99 55.85(3.67) — — CycleGAN 45 degree Cityscapes 0.99 0.69 54.94(5.20) — —

Experiments

Settings. We test three widely adopted generative models, DCGAN, PGAN, and Cycle-GAN, and three datasets: MNIST, CelebA and Cityscape. See below for details on GAN settings and dataset descriptions. For the root models, we train DCGANs from scratch on MNIST and CelebA, and use pre-trained PGAN and CycleGAN.

We answer the following questions empirically through experiments.

Can decentralized attributability be achieved through orthogonal keys? For each dataset, we compute twenty keys (Eq. (8)) and their corresponding generative models (Eq. (9)). Table 1 reports the empirical averaged distinguishability and attributability for the collections. For comparison, we randomly sample 20 data-compliant keys by solving an alternative to Eq. (8) where the angle between keys is constrained to 45 deg. The results are presented in the same table. Generation quality metrics (∥Δx∥ and FID) are reported in the same table.

Is there a limited capacity of keys? For real-world applications, we would need the capacity of keys to achieve decentralized attribution to be large. From the analysis, the capacity is limited by the availability of orthogonal keys, which is required by attribution, and the generation quality. In FIG. 3A, we report the quantities for 1500 keys generated for MNIST: Orthogonality o_(i)=Σ_(j=1) ^(j−1)|ϕ_(j) ^(T)ϕ_(i)|/(i−1)(o_(i)=0), key-perturbation correlation c_(i)−ϕ_(i) ^(T)Mϕ_(i), d_(max)(ϕi), distinguishability D(G_(ϕ) _(i) ), attributability A({G_(ϕ) _(j) }_(j−1) ^(i)), and generation quality for i=1, . . . , 1500. Some remarks: (1) Nearly orthogonal keys abound due to the high-dimensionality of the output space, for which decentralized attribution is achieved. (2) Larger ci indicates more involvement of the key in the eigenspace of M with large eigenvalues. There is a positive correlation (0.63) between c and the FID scores, as expected. (3) d_(max) is bounded and so is ∥Δx∥. Samples from the generator with the largest ∥Δx∥ are illustrated in FIG. 3A. The results suggest that the registry can use c and d_(max) to monitor the generation quality.

Approximation of M: Since the computation of M (thus c) is expensive for deep generative models with high-dimensional outputs, we seek an empirical approximation of M. Our hypothesis is that the structured patterns associated with eigenvectors of large eigenvalues are mostly associated with in the sensitivities with respect to parameters from the later layers of the generators, and therefore we can approximate M using part of the Jacobian with respect to only those layers. To test the hypothesis, we train relatively shallow DCGANs for MNIST and CelebA, and compute the cosine similarities between the eigenvectors of M with the largest eigenvalue and those from the approximations of M using the last two layers. Results are presented in FIG. 3B, and suggest that it is viable to approximate the largest eigenvectors using the last layers.

How do post-processes affect attributability and generation quality? We consider five types of post-processes: blurring, cropping, noise, JPEG conversion and the combination of these four, and assume that the post-processes are known by the model publishers who then improve the robustness of decentralized attribution by incorporating these processes as differentiable layers and solving Eq. (10). Examples of the post-processed images from non-robust and robust generators are compared in FIGS. 4A-4F. Implementation: Blurring uses Gaussian kernel width uniformly drawn from ₃ ¹{1.3.5.7.9}. Cropping crops images with uniformly picked ratio between 80% and 100% and scales the cropped images back to the original size using bilinear interpolation. Noise adds iid white noise with standard deviation uniformly drawn in [0, 0.3]. JPEG applies JPEG compression. Combination performs each attack with a 50% chance in the order of Blurring, Cropping, Noise and JPEG. We use implementations for differentiable blurring and JPEG. For robust training against each post-process, we apply the post-process to mini-batches with 50% probability. Results: We report in Table 2 the attributability before and after robust training of distinguishability. Blurring, Cropping and Combination are all effective before robust training. Defense against these random post-processes can be achieved except for Combination. Table 3 reports ∥Δx∥ and FID scores of the robust models, showing the trade-off between attributability and generation quality. See below for more results of robust training.

Related Work

Fingerprints of GANs. Researches have shown that convolutional neural network based generator leaves artifacts. Marra et. al. empirically showed that the artifact can be used as a fingerprint.

TABLE 2 Distinguishability (top), attributability (btm) before (Bfr) and after (Aft) robust training. GANs Dataset Blurring Cropping Noise JPEG Combination — — Bfr Aft Bfr Aft Bfr Aft Bfr Aft Bfr Aft DCGAN MNIST 0.49 0.96 0.52 0.99 0.85 0.99 0.54 0.99 0.50 0.66 DCGAN CelebA 0.49 0.99 0.49 0.99 0.95 0.98 0.51 0.99 0.50 0.85 PGAN CelebA 0.50 0.98 0.51 0.99 0.97 0.99 0.96 0.99 0.50 0.76 CycleGAN Cityscapes 0.49 0.92 0.49 0.87 0.98 0.99 0.55 0.99 0.49 0.67 DCGAN MNIST 0.49 0.96 0.49 0.97 0.85 0.98 0.53 0.99 0.49 0.65 DCGAN CelebA 0.50 0.99 0.50 0.99 0.95 0.99 0.51 0.99 0.50 0.85 PGAN CelebA 0.50 0.97 0.50 0.99 0.96 0.98 0.96 0.99 0.50 0.76 CycleGAN Cityscapes 0.50 0.92 0.50 0.86 0.97 0.98 0.54 0.99 0.50 0.67

TABLE 3 ∥Δ.r∥ (top) and FID score (btm) w/ and w/o robust training. Standard deviations in parenthesis. DCGAN-M: DCGAN for MNIST, DCGAN-C: DCGAN for CelebA. FID score not applicable to CycleGAN. Lower is better. GANs Non-robust Blurring Cropping Noise JPEG Combination DCGAN-M 5.20(0.31) 15.96(2.18) 9.17(0.65) 5.93(0.34) 6.48(0.94) 17.08(1.86) DCGAN-C 4.19(0.18) 11.83(0.65) 9.30(0.31) 4.75(0.17) 6.01(0.29) 13.69(0.59) PGAN 9.29(0.95) 18.49(2.04) 21.27(0.81) 10.20(0.81) 10.08(1.03) 24.82(2.33) CycleGAN 55.85(3.67) 68.03(3.62) 80.03(3.59) 55.47(1.60) 57.42(2.00) 83.94(4.66) DCGAN-M 5.68(0.23) 41.11(20.43) 21.58(2.44) 5.79(0.19) 6.50(1.70) 68.16(24.67) DCGAN-C 52.09(2.20) 73.62(6.70) 98.86(9.51) 59.51(1.60) 60.35(2.57) 87.29(9.29) PGAN 21.62(1.73) 28.15(3.43) 47.94(5.71) 25.43(2.19) 22.86(2.06) 45.16(7.87)

However, their method depends on the dissimilarities of the target data. Yu et al. trained external classifier to identify the images from a finite and fixed set of generators, and showed that the classifier can achieve robustness against post-processed images by fine-tuning the classifier using post-processed images. But the result is not guaranteed to have the same performance when the set of generators grows arbitrarily. Albright et al. showed that they can find the origin of images by solving the generator inversion problem. This method requires that the registry save all generators. Furthermore, the registry needs to solve the optimization problem for all generators.

Digital watermarking. Digital watermarking has been used for identifying the ownership of digital signals. Research on watermarking focused on the least significant bits in images and frequency domain. Zhu et al. showed that GANs can be used for watermarking by introducing various operation layers to the training step. Since watermarks are directly added to the outputs, they are similar to the presented toy case. Along the same direction, Fan et al. imposed passport to classification networks. Without proper passport, the classification accuracy of the network drops. Their approach, however, has not been extended to the decentralized attribution setting.

Conclusion

This paper investigated the feasibility of decentralized attribution for generative models. We used a protocol where a registry generates and distributes keys to users, and the user creates a key-dependent generative model for which the outputs can be correctly attributed by the registry. Our investigation led to simple design rules of the keys to achieve correct attribution while maintaining reasonable generation quality. Specifically, correct attribution requires keys to be data compliant and orthogonal; and generation quality can be monitored through data- and model-dependent metrics. With concerns about adversarial post-processes, we empirically show that robust attribution can be achieved with further loss of generation quality. This study defines the design requirements for future protocols for the creation and distribution of attributable generative models.

Broader Impact

With recent advances of generative models, researchers focus on the potential misuses and their forensics. Current state-of-the-art models can generate realistic fake images, voices and videos. Against these developments, studies of forensic have also been in the spotlight. This paper takes a different perspective than this ongoing competition between the two sides. We are motivated by the requirement of model attribution, i.e., the ability to tell which exact models do the contents come from, in addition to whether the contents are machine generated or not.

To this end, the paper focused on a regulation approach in the setting where generative models are white-box to end users, keys are black-box (withheld by the model publishers), and datasets are proprietary. While we focus on the technical feasibility of decentralized attribution of generative models, the applicability of the proposed method would require discussions beyond the scope of the paper. We assume that the protocol, i.e., key distribution by the model publisher and key-dependent training on the user end, can be embraced by all stakeholders involved (e.g., social media platforms and news organizations). While this protocol does not eliminate risks from individual adversaries, it will be a necessary constraint on publishers that have the computational, technological, and data resources to create and distribute high-impact machine-generated contents.

Proof of Proposition 1

Proposition 1. For the toy case, if

ϵ > max_(x ∼ P_(D)){x_(∝)}, D(G_(ϕ)) − 1

for all ϕ⊂Φ and A(

)−1.

Proof. Let φ and φ′ be any pair of keys such that ϕ^(T)ϕ′=0, and let x, x′, and x₀ be sampled from P_(G) _(ϕ) , P_(G) _(ϕ′) , and

, respectively. When ε>max_(x˜)

{∥x∥_(∝)}, we have

$\begin{matrix} {{{\text{?}\text{?}} - {\text{?}\left( {x_{0} + {\text{?}\text{?}}} \right)}} = {{{\text{?}x_{0}} + \text{?}} > {{\text{?}x_{0}} + {\text{?}\left\{ {{\left. {x\text{?}} \right\} > {{\text{?}x_{0}} - {\text{?}x_{0}}}} = {0.\text{?}\text{indicates text missing or illegible when filed}}}\mspace{191mu} \right.}}}} & (11) \end{matrix}$

Combined with the data-compliant assumption ϕ^(T)x₀<0, we have D(G_(ϕ))−1. Further, since

ϕ^(T) x′=ϕ ^(T)(x ₀+εϕ′)=ϕ^(T) x ₀<0.  (12)

we have A(

)=1.

Empirical Test for the Linear Approximation

For first-order analyses, we approximate the key-dependent generative model to be updated from the root model through θ=θ₀−Δθ, where

Δθ = γ ⁢ x ∼ P G D ⁡ [ ∇ θ ⁢ x T θ 0 ] ⁢ ϕ . ( 13 ) and x - x 0 ⁢  ∇ θ ⁢ x 0  θ 0 ⁢ Δ0 . ( 14 )

Let J(x)=∇_(θ)x|_(θ) ₀ and

M = 𝔼_(?)[J(x)]𝔼_(?)[J(x)^(T)] ?indicates text missing or illegible when filed                    

We focus on testing the following result of the linear approximation: For ϕ and G_(ϕ) with high distinguishability, we should observe that with high probability,

$\begin{matrix} {{{{{\phi^{T}x} - {\phi^{T}\left( x_{0} \middle| {\text{?}{J\left( x_{0} \right)}{{\mathbb{E}}_{\text{?}}\left\lbrack {J(x)}^{T} \right\rbrack}\phi} \right)}} > 0.}{\text{?}\text{indicates text missing or illegible when filed}}}\mspace{275mu}} & (15) \end{matrix}$

for x₀˜P_(G) _(ϕ) . To test this, we use a DCGAN trained on MNIST as G₀. We train 20 keys and update Gs correspondingly following the method detailed in the Experiments section. The resulting average distinguishability from the 20 generative models is 0.99.

To compute Pr(ϕ^(T){tilde over (x)}>0), we calculate J(x₀) and

x ∼ P G D ⁡ [ J ⁡ ( x ) ]

based on samples from G₀. From Eq. (13),

${{{{\Delta\theta}{ - }{{\gamma\mathbb{E}}_{\text{?}}\left\lbrack {J(x)}^{T} \right\rbrack}\phi}} - {\text{?}{\sqrt{\phi^{T}M\;\phi}.\text{?}}\text{indicates text missing or illegible when filed}}}\mspace{301mu}$

Therefore γ=∥Δθ∥/√{square root over (ϕ^(T)Mϕ)}. ∥Δθγ can be directly computed by comparing θ and θ₀; M can be computed through SVD on

? ⁡ [ J ⁡ ( x ) ] ?indicates text missing or illegible when filed                    

(the tested DCGAN has 1,065,984 parameters, and output dimension of 1024, thus J∈

^(1021×1.065.984)). Empirical test shows ro ₂₀ ¹Σ_(o∈Φ)Pr(ϕ^(T)x>0)−0%. Empirical test for Conjecture 1

Conjecture 1. Let the (i,j)th element of

Σ ⁡ ( x ) = J ⁡ ( x ) ⁢ ? ⁡ [ J ⁡ ( x ) ] T - M ?indicates text missing or illegible when filed                    

be Σ_((i,j)) a with variance σ_(ij) ². Then Σ_((i,j)) is approximately drawn i.i.d. from

(0, σ_(ij) ²).

Normality. We use a DCGAN trained on MNIST as G₀ and collect 512 samples of Σ by sampling x₀˜P_(G) _(o) . We empirically pick the best distributions for Σ_((i,j)). To do that, we calculate the Akaike Information Criterion (AIC) and the Bayesian Information Criterion (BIC) for each Σ_((i,j)) (10242 calculations in total). Candidate distributions include beta, bimbaumsaunders, exponential, extreme value, gamma, generalized extreme value, generalized pareto, inversegaussian, logistic, loglogistic, lognormal, nakagami, normal, rician, tiocationscale, and weibull distributions. We only report AIC and BIC of normal and extreme value distributions. Among all, the lowest mean AIC and BIC are found from the normal distribution (AIC=−26.51 and BIC=−18.03). The second best comes from the extreme value distribution (AIC=161.42 and BIC=169.90). From the reported results, we argue that it is reasonable to assume normality for

Independence. Due to normality, we test independence through correlations. In theory, this requires a 10242-by-10242 covariance matrix for all Σ_((i,j)). Without overloading the computational resources, we randomly pick one elements from Σ_((i,j)) and compute correlation coefficient with others (1024² calculation). We do such calculation for fifty times without duplication. The resulting average absolute value of the correlations is smaller than 0.1, suggesting that the independence assumption is reasonable. Multiple repetition of calculations did not show notable variations of correlations.

Empirical test for Conjecture 2

Conjecture 2. Denote Λ−{λ₁ . . . λ_(d) _(x) } as the eigenvalues of M. For existing deep generative models, there exists and only exists a subset of eigenvalues that are strictly positive.

We use the same DCGAN trained on MNIST and CelebA as the root models to compute

x ∼ P G D ⁡ [ J ] .

SVD on the resulting matrix reveals the eigenvalues of M, which are reported in FIG. 5.

Proof of Theorem 1

Theorem 1. Let

d_(max)(ϕ) = max_(x ∼ P_(P))ϕ^(T)x, σ²(ϕ) = Σ_(i, j)σ_(i, j)²ϕ_(i)²ϕ_((j))²,

and δ_(d) be a positive number greater than (

). For the realistic case and for a given key ϕ∈Ω, if

$\begin{matrix} {{{{{\Delta\text{?}\left( \text{?} \right)}} \geq {\begin{matrix} {{d_{\max}\left( \text{?} \right)}\sqrt{\text{?}M^{2}\text{?}}} \\ {{\text{?}M\text{?}} - {{\sigma\left( \text{?} \right)}\sqrt{\log\left( \frac{1}{\text{?}} \right)}}} \end{matrix}.{D\left( \text{?} \right)}} \geq {1\mspace{14mu}{\delta_{d}/2}}}{\text{?}\text{indicates text missing or illegible when filed}}}\mspace{259mu}} & (16) \end{matrix}$

Proof. We first note that due to data compliance of keys,

,

(ϕ^(T)x<0)┘=1. Therefore

D(G_(?)) ≥ 1 − δ_(d)/2  iff  E_(?)[(?^(T)x > 0) ≥ 1 − δ_(d); ?indicates text missing or illegible when filed                 

i.e., Pr(ϕ^(T)x>0)≥1−δ_(d) for x˜P_(G) _(ϕ) . We now seek a key-dependent lower bound on ε to satisfy this inequality. We first connect generation quality to the step size (learning rate) γ following the linear approximation:

∥Δx(ϕ)|=∥γMϕ∥=γ√{square root over (ϕ^(T) M ²ϕ)}.  (17)

Next, given φ, we look for a sufficiently large γ, so that ϕ^(T)x>0 with probability at least 1−δ_(d). To do so, let x and x₀ be sampled from P_(G) _(ϕ) and P_(G) _(o) , respectively. Then with first order approximations we have

$\begin{matrix} {{{\text{?}\text{?}} = {{\text{?}\left( {x_{0} + {\text{?}{J\left( x_{0} \right)}{\text{?}\left\lbrack {J\left( \text{?} \right)}^{T} \right\rbrack}\text{?}}} \right)} = {{\text{?}x_{0}} + {\text{?}\text{?}M\text{?}} - {\text{?}\text{?}{\sum{{\text{?}.\text{?}}\text{indicates text missing or illegible when filed}}}}}}}\mspace{200mu}} & (18) \end{matrix}$

For Pr(ϕ^(T)x>0)≥1−δ_(d), γ should satisfy

Pr(ϕ^(T)Σϕ>−ϕ^(T) x ₀/γ−ϕ^(T) Mϕ)≥1−δ_(d).  (19)

Since d_(max)(ϕ)≥−ϕ^(T)x₀; it is sufficient to have

Pr(ϕ^(T) Σϕ>−d _(max)(ϕ)/γ−ϕ^(T) Mϕ)≥1 δ_(d).  (20)

From Conjecture 1, ϕ^(T)Σϕ˜

(0, σ²(ϕ)). Due to the symmetry of p(ϕ^(T)Σϕ), the sufficient condition for γ in Eq. (20) can be rewritten as

Pr(ϕ^(T)Σϕ≤ϕ^(T) Mϕd _(max)(ϕ)/γ)≥1 δ_(d).  (21)

Recall the following tail bound of x˜

(0, σ²) for y≥0:

Pr(x<σy)>1−exp(−y ²/2).  (22)

Compare Eq. (22) with Eq. (21), the sufficient condition becomes

$\begin{matrix} {\left. {{{\text{?}M\text{?}} - {{d_{\max}(\phi)}/\text{?}}} \geq {{\sigma\left( \text{?} \right)}\sqrt{\log\left( \frac{1}{\delta_{d}^{2}} \right)}}}\Rightarrow{\text{?} \geq {{\frac{d_{\max}\left( \text{?} \right)}{{\text{?}M\text{?}} - {{\sigma\left( \text{?} \right)}\sqrt{\log\left( \frac{1}{\text{?}} \right)}}}.\text{?}}\text{indicates text missing or illegible when filed}}} \right.\mspace{265mu}} & (23) \end{matrix}$

Using Eq. (17), we have

$\begin{matrix} {{{{\Delta\text{?}\left( \text{?} \right)}} \geq {{\begin{matrix} {{d_{\max}\left( \text{?} \right)}\sqrt{\text{?}M^{2}\text{?}}} \\ {{\text{?}M\text{?}} - {{\sigma\left( \text{?} \right)}\sqrt{\log\left( \frac{1}{\text{?}} \right)}}} \end{matrix}.\text{?}}\text{indicates text missing or illegible when filed}}}\mspace{239mu}} & (24) \end{matrix}$

provided that

$\begin{matrix} {{{{{\text{?}M\text{?}} - {{\sigma\left( \text{?} \right)}\sqrt{\log\left( \frac{1}{\text{?}} \right)}}} > {0\mspace{14mu}{or}}}{\delta_{d} > {{{\exp\left( {{- \frac{1}{2}}\left( \frac{\text{?}M\text{?}}{\sigma\left( \text{?} \right)} \right)^{2}} \right)}.\text{?}}\text{indicates text missing or illegible when filed}}}}\mspace{259mu}} & (25) \end{matrix}$

Empirical Test for Conjecture 3

Conjecture 3. |Δx∥≤τd_(max).

The conjecture comes from the following approximations: First, from Conjecture 1, we observe that {σ_(ij)}² are small. Using the proof of Theorem 1, a sufficient degradation of generation quality can be approximated by

$\begin{matrix} {\mspace{79mu}{{{\Delta\text{?}(\phi)}} \approx {{\begin{matrix} {{{d_{\max}(\phi)}\sqrt{\phi^{T}M^{2}\phi}} -} & {d_{\max}\sqrt{c^{T}\Lambda^{2}c}} \\ {\phi^{T}M\;\phi} & {c^{T}\Lambda\; c} \end{matrix}.\text{?}}\text{indicates text missing or illegible when filed}}}} & (26) \end{matrix}$

where c=P^(T)ϕ and M=PAP^(T). From Lemma 1,

$\begin{matrix} {\mspace{79mu}{\frac{\sqrt{c^{T}\Lambda^{2}c}}{c^{T}\Lambda\; c} \in {\left\lbrack {1 \cdot \frac{1\text{?}{\lambda_{\max}/\lambda_{\min}}}{2\sqrt{\lambda_{\max}/\lambda_{\min}}}} \right\rbrack.\mspace{20mu}{Let}}}} & (27) \\ {\mspace{79mu}{\tau = {{\frac{1 + {\lambda_{\max}/\lambda_{\min}}}{2\sqrt{\lambda_{\max}/\lambda_{\min}}}.\text{?}}\text{indicates text missing or illegible when filed}}}} & (28) \end{matrix}$

then |Δx∥<τd_(max).

Useful Lemmas

Lemma 1 is used for Conjecture 3 and Lemmas 2 for the proof of Theorem 2.

Lemma 1. Let c∈

^(n) and ∥c∥=1, Λ=diag(λ1 . . . λ_(n)) be positive definite. Then

$\begin{matrix} {\begin{matrix} \sqrt{c^{T}\Lambda^{2}c} \\ {c^{T}\Lambda\; c} \end{matrix}{{C\;\left\lbrack {1.\begin{matrix} {1 + {\lambda_{\max}/\lambda_{\min}}} \\ {2{\left. \sqrt{}\lambda_{\max} \right./\lambda_{\min}}} \end{matrix}} \right\rbrack}.}} & (29) \end{matrix}$

Proof. Let x=:[c₁ ² . . . c_(n) ²:, a=:λ₁ ² . . . λ_(n) ²], and b=[λ₁ . . . λ_(n)]. Then c^(T)Λ²c=a^(T)x and c^(T)Λc=b^(T)x.

We now consider the following problem:

$\begin{matrix} {\mspace{79mu}{{{\max\limits_{\text{?}}{\frac{1}{2}\log\mspace{11mu} a^{T}\text{?}}} - {\log\mspace{11mu} b^{T}\text{?}}}\mspace{20mu}{{{s.t.\mspace{14mu} 1^{T}}\text{?}} = 1}\mspace{20mu}{x_{i} \geq {0.\mspace{11mu}\text{∀}{i.\text{?}}\text{indicates text missing or illegible when filed}}}}} & (30) \end{matrix}$

The KKT conditions for this problem are

$\begin{matrix} {\mspace{79mu}{{{{{- \frac{1}{2a^{T}\text{?}}}a} + {\frac{1}{b^{T}\text{?}}b} + {\lambda 1} - \mu} = 0.}\mspace{20mu}{{1^{T}\text{?}} - 1}{\text{?} \geq {0.\mu_{i}} \geq {0.\mspace{11mu}\text{∀}i}}\mspace{20mu}{{\mu^{T}\text{?}} - 0.}{\text{?}\text{indicates text missing or illegible when filed}}}} & (31) \end{matrix}$

where λ and μ are the Lagrangian multipliers.

When b has unique elements, there exist two sets of KKT points: x is either one-hot, or x has zero entries except for elements i and j where x_(i)=b_(j)/(b_(i)+b_(j)) and x_(j)=b_(i)/(b_(i)+b_(j)), for all (i, j) combinations. If b has repeated elements, then we can combine these elements and reach the same conclusion.

When x is one-hot, the objective is log a_(i)/2−log b_(i)=0. For the second type of solutions and let τ_(ij)−λ_(i)/λ_(j), we have

$\begin{matrix} {\begin{matrix} {\mspace{79mu}{{{\frac{1}{2}\log\mspace{11mu} a^{T}\text{?}} - {\log\mspace{11mu} b^{T}\text{?}}} = {{\frac{1}{2}\log\mspace{11mu}\frac{{a_{i}b_{j}} + {a_{j}b_{i}}}{b_{i} + b_{j}}} - {\log\mspace{11mu}\frac{2b_{i}b_{j}}{b_{i} + b_{j}}}}}} \\ {= {{\frac{1}{2}\log\mspace{14mu}\lambda_{i}\lambda_{j}} - {\log\mspace{11mu}\frac{2\lambda_{i}\lambda_{j}}{\lambda_{i} + \lambda_{j}}}}} \\ {= {{\log\mspace{11mu}\frac{1 + \tau_{ij}}{2\left. \sqrt{}\tau_{ij} \right.}} \geq 0.}} \end{matrix}{\text{?}\text{indicates text missing or illegible when filed}}} & (32) \end{matrix}$

where equality holds when τ_(ij)−1. Since the objective monotonically increases with respect to τ_(ij)>1, the maximum is reached when τ_(ij)−λ_(max)/λm_(in).

Lemma 2. Let a,b∈

^(n)n∥a∥=1, ∥b∥=1, and a^(T)b≤0. Let V∈

^(n×n). Then max_(a){a^(T)V^(b)}=√b^(T)V^(T)V^(b)−(b^(T)V^(b))².

Proof. Consider the following problem

$\begin{matrix} {\begin{matrix} {\mspace{79mu}\min\limits_{\text{?}}} & {{- a^{T}}{Vb}} \\ {\mspace{76mu}{s.t.}} & {{a^{T}b} \leq 0} \\ \; & {{a^{T}a} = 1.} \end{matrix}{\text{?}\text{indicates text missing or illegible when filed}}} & (33) \end{matrix}$

with the following KKT conditions:

−Vb+μb+2λa=0

a ^(T) b≤0

a ^(T) a−1.  (34)

The solution is

$\begin{matrix} \begin{matrix} {\lambda - {a^{T}{Vb}\text{/}2}} \\ {\mu - {b^{T}{Vb}}} \\ {a = {\frac{\left( {V - {b^{T}{V{bI}}}} \right)b}{\sqrt{{b^{T}V^{T}{Vb}} - \left( {b^{T}{Vb}} \right)^{2}}}.}} \end{matrix} & (35) \end{matrix}$

Note that

$\begin{matrix} \begin{matrix} {{{\left( {V - {b^{T}{V{bI}}}} \right)b}}^{2} = {{b^{T}\left( {V^{T} - {b^{T}{V{bI}}}} \right)}\left( {V - {b^{T}{V{bI}}}} \right)b}} \\ {= {{b^{T}V^{T}{Vb}} - {\left( {b^{T}{Vb}} \right)^{2}.}}} \end{matrix} & (36) \end{matrix}$

thus b^(T)V^(T)Vb−(b^(T)Vb)²≥0.

Since the Hessian of the Lagrangian with respect to a is 2λI. and from the solution

$\begin{matrix} \begin{matrix} {\lambda = {a^{T}{Vb}\text{/}2}} \\ {= {{\sqrt{{b^{T}V^{T}{Vb}} - \left( {b^{T}{Vb}} \right)^{2}}\text{/}2} \geq 0.}} \end{matrix} & (37) \end{matrix}$

therefore the solution is the minimizer, i.e., max_(a){a^(T)Vb}−√{square root over (b^(T)V^(T)Vb (b^(T)Vb)²)}.

Proof of Theorem 2

Theorem 2. Let

$d_{\min}^{*} - {\min_{{\phi\epsilon\Phi},{x \sim P_{D}}}{{\phi^{T},x,{{{\overset{\_}{\sigma}}^{2}(\phi)} - \sqrt{\phi^{T}V^{T}{V\phi}\mspace{14mu}\left( {\phi^{T}{V\phi}} \right)^{2}}}}}}$

, and V_((i,j))−σ_(ij) ². When D(G)≥1−δ_(d) for all Gϕ∈

, if the degradation of generation quality for all models in

satisfies

$\begin{matrix} {\mspace{76mu}{{{\Delta\text{?}(\phi)}} < {{\frac{d_{\min}^{*}\left. \sqrt{}\phi^{T} \right.M^{2}\phi}{\sqrt{{\phi^{T}M^{2}\phi} - \left( {\phi^{T}M\;\phi} \right)^{2}} + {{\sigma(\phi)}\sqrt{\log\mspace{11mu}\left( \frac{1}{\text{?}} \right)}}}.\text{?}}\text{indicates text missing or illegible when filed}}}} & (38) \end{matrix}$

and ϕ^(T)ϕ′ for all ϕ, ϕ′∈Ω, then A(

)≥1−(δ_(d)+δ_(a))/2.

Proof. Let ϕ and ϕ′ be any of the two orthogonal keys, and x′ and x₀ be sampled from P_(G) _(ϕ′) and P_(G) ₀ , respectively. A(

)≥1 (δ_(d)|δ_(a))/2 and D(G)≥1 δ_(d), for all G∈

together imply that Pr(ϕ^(T)x′<0)≥1−δ_(a). Now we focus on deriving the sufficient conditions for this inequality. From first order approximations,

$\begin{matrix} {\begin{matrix} {\mspace{79mu}{{\phi^{T}\text{?}} = {\phi^{T}\left( {\text{?} + {\text{?}\left( \phi^{\prime} \right){J\left( \text{?} \right)}{\mathbb{E}}{\text{?}\left\lbrack {J\left( \text{?} \right)}^{T} \right\rbrack}\phi^{\prime}}} \right)}}} \\ {= {{\phi^{T}\text{?}} + {\text{?}\left( \phi^{\prime} \right)\phi^{T}M\;\phi} + {\text{?}(\phi)\phi^{T}{\sum{\phi.}}}}} \end{matrix}{\text{?}\text{indicates text missing or illegible when filed}}} & (39) \end{matrix}$

Therefore

Pr(ϕ^(T) x′<0)−Pr(ϕ^(T)Σϕ′<ϕ^(T) x ₀/γ(ϕ′)ϕ^(T) Mϕ′)≥Pr(ϕ^(T) Σϕ′<d _(min)(ϕ)/γ(ϕ′)−ϕ^(T) Mϕ′).   (40)

Note that the RHS of Eq. (40) suggests that γ(ϕ′) needs to be sufficiently small for Pr(ϕ^(T)x′<0) to be large. To see where that upper bound is, we start by noting that ϕ^(T)Σϕ′ has zero mean and is normally distributed. To analyze its variance, we use Lemma 2 to show that

Var(ϕ^(T)Σϕ′)≤σ²(ϕ′)=√ϕ′^(T) V ^(T) Vϕ′−(ϕ′^(T) Vϕ′)².  (41)

where V_((i,j))=σ_(ij) ²,

Using the same tail bound of normal distribution as in Theorem 1, γ(ϕ′) is sufficient small if

$\begin{matrix} {\mspace{79mu}\left. {{{d_{\min}\left( {\phi/\text{?}} \right)}\left( \phi^{\prime} \right)\mspace{11mu}\phi^{T}M\;\phi^{\prime}} \geq {{\overset{\_}{\sigma}\left( o^{\prime} \right)}\sqrt{\log\begin{pmatrix} 1 \\ {\delta\text{?}} \end{pmatrix}}}}\Rightarrow{{\text{?}\left( o^{\prime} \right)} \leq \left\{ {{\begin{matrix} \frac{d_{\min{(o)}}}{o^{T}{{Mo}^{\prime} \cdot \text{?}}\left( \phi^{\prime} \right)\sqrt{\log\mspace{11mu}\left( \frac{1}{\text{?}} \right)}} & {{{if}\mspace{14mu}\phi^{T}M\;\phi^{\prime}} + {{\sigma\left( \phi^{\prime} \right)}\sqrt{{\log\left( \frac{1}{\text{?}} \right)} > 0}}} \\ {+ \propto} & {otherwise} \end{matrix}.\text{?}}\text{indicates text missing or illegible when filed}} \right.} \right.} & (42) \end{matrix}$

Since ∥Δx(ϕ′)∥=γ(ϕ′)√{square root over (ϕ′^(T)M²ϕ′)}, we have

$\begin{matrix} {{{\Delta\text{?}\left( o^{\prime} \right)}} \leq \left\{ {\begin{matrix} \frac{\text{?}\sqrt{{o^{\prime}}^{T}M\text{?}}}{o^{T}{Mo}^{\prime}\text{?}\sqrt{\log\left( \text{?} \right)}} & {{{if}\mspace{14mu} o^{T}{Mo}^{\prime}} -} \\ {+ \propto} & {otherwise} \end{matrix}{\sigma\left( o^{\prime} \right)}{\sqrt{{\log\left( \frac{1}{\text{?}} \right)} > 0}.\text{?}}\text{indicates text missing or illegible when filed}} \right.} & (43) \end{matrix}$

We would like to find a lower bound of the RHS of Eq. (43) that is independent of ϕ≠ϕ′. To this end, first denote d_(min)′=min_(ϕ)d_(min)(ϕ). Now use Lemma 2 again to derive an upper bound of ϕ^(T)Mϕ′:

ϕ^(T) Mϕ<√{square root over (ϕ′^(T) M ²ϕ−(ϕ′^(T) Mϕ)²)}.  (44)

Replace ϕ^(T)Mϕ′ in Eq. (43) with its upper bound to reach a ϕ-independent sufficient condition for |Δx(ϕ′)|:

$\begin{matrix} {\mspace{79mu}{{{\Delta\text{?}\left( o^{\prime} \right)}} < {{\frac{d_{\min}\left. \sqrt{}{o^{\prime}}^{T} \right.M^{2}\text{?}}{\sqrt{{{o^{\prime}}^{T}M^{2}o^{\prime}} - \left( {{o^{\prime}}^{T}{Mo}^{\prime}} \right)^{2}} - {{\sigma\left( o^{\prime} \right)}\sqrt{\log\left( \frac{1}{\text{?}} \right)}}}.\text{?}}\text{indicates text missing or illegible when filed}}}} & (45) \end{matrix}$

Limited Capacity of Keys

We generate 1500 keys for MNIST: orthogonality o_(i)=Σ′_(j−1) ¹|ϕ_(J) ^(T)ϕ_(i)/(i−1) (ϕI=0), key-perturbation correlation c_(i)−ϕ_(i) ^(T)Mϕ_(e), d_(max)(ϕ_(i)), distinguishability D(G_(ϕ) _(i) ), attributability A({G_(ϕ) _(i) }_(j−1) ^(i)), lack of generation quality |Δx∥ and FID for i=1 . . . 1500. Some remarks: (1) d_(max) is bounded and so is |Δx∥ (FIG. 6A). (2) Larger c_(i) indicates more involvement of the key in the eigenspace of M with large eigenvalues. There is a positive correlation (0.63) between c and the FID scores, as expected (FIG. 6B). (3) Nearly orthogonal keys abound due to the high-dimensionality of the output space, for which decentralized attribution is achieved (FIG. 6B). Thus, the results suggest that the registry can use c and d_(max) to monitor the generation quality.

Approximation of M: The hypothesis is that the structured pattern of large eigenvectors is associated with eigenvectors of the later layers of the generators. Therefore, M can be approximated using the Jacobian of these layers. For empirical experiments, we train four-layer DCGANs for MNIST and CelebA, and compute the cosine similarities between the largest eigenvector of M and the largest eigenvectors of Jacobian of each of layers. Results are presented in FIGS. 7A and 7B with visual examples. Also, it is viable to approximate the largest eigenvectors with the last layers.

Examples of GANs

In the paper, we show examples from PGAN with CelebA. Here, we illustrate other GANs examples. For FIGS. 8A-8F, 9A-9F, and 10A-10F, (a) 1st-2rd row: authentic data, samples from the non-robust generator (b-f) 1st-2rd rows: worst-case post-process, samples from robust training against the specific post-processes (before the post-processes). 3rd row for all: numerical differences between 2nd row of (a) and 2nd row of each case. Thus, the differences show the effect of robust training on attribution.

Training Details K.1 Parameters

We adopt Adam optimizer for gradient descent. We attach other parameters in Table 4. Note that we fix the hyper-parameters when we optimize Eq. (Robust training) in Implementation.

K.2 Training Time

For experimental validations, we use V:100 Tesla GPUs. Exact number of GPUs are reported in Table 5.

TABLE 4 Hyper-parameters for training Eq. (Key generation) (top) and Eq. (Generative models) (btm). Equations are in Implementation. Batch Learning GANs Dataset Size Rate β₁ β₂ Epochs C DCGAN MNIST 128 0.001 0.5 0.99 10 — DCGAN CelebA 64 0.001 0.5 0.99 10 — PGAN CelebA 32 0.00 0.5 0.99 10 — CycleGAN Cityscapes 4 0.001 0.5 0.99 20 — DCGAN MNIST 16 0.0005 0.5 0.99 10 10 DCGAN CelebA 64 0.0005 0.5 0.99 10 10 PGAN CelebA 16 0.0005 0.0 0.99 5 100 CycleGAN Cityscapes 1 0.0005 0.5 0.99 5 1000

TABLE 5 Training time (in minute) of one key (Eq. (Key generation)) and one generator (Eq. (Generative models)). DCGAN-M: DCGAN for MNIST, DCGAN-C: DCGAN for CelebA. Equations are in Implementation. GANs CPUs Key Naive Blurring Cropping Noise JPEG Combination DCGAN-M 1 1.77 8.52 4.12 3.96 4.19 5.71 5.12 DCGAN-C 1 5.31 9.12 10.33 9.56 10.35 10.25 10.76 PGAN 2 50.89 141.07 140.05 131.90 133.46 132.46 135.07 CycleGAN 1 20.88 16.04 16.26 15.43 15.71 15.98 16.41

Ablation Study

We attach the table of ablation study of how C affects the result of distinguishability, attributability, ∥Δx∥ and FID scores in Table 6. C does not affect to the distinguishability and attributability. But C improves ∥Δx∥ and FID for every generators. Furthermore, we investigate how C term affects the robustness in Table 7 and Table 8. We can observe that, as C increases, robustness decreases but generation quality increases.

TABLE 6 Empirical averaged distinguishability (D), attributability (A ( 

 )). ∥Δ.r∥ and FID scores. Standard deviations reported when applicable, or omitted if ≤0.05. FID of ^(G) ⁰ ^(.) (FID₀) is the baseline. FID is not applicable to CycleGAN. GANs Dataset C D A( 

 ) ∥Δ.r∥ FID₀ FID DCGAN MNIST  10 0.99 0.99 5.20(0.31) 4.98(0.15) 5.68(0.23) DCGAN MNIST 100 0.99 0.99 4.09(0.53) — 5.32(0.11) DCGAN MNIST  1K 0.99 0.99 3.88(0.60) — 5.23(0.12) DCGAN CelebA  10 0.99 0.99 4.19(0.18) 33.95(0.13) 52.09(2.20) DCGAN CelebA 100 0.99 0.99 3.08(0.27) — 45.02(3.37) DCGAN CelebA  1K 0.99 0.99 2.55(0.36) — 40.85(3.41) PGAN CelebA 100 0.99 0.99 9.29(0.95) 13.31(0.07) 21.62(1.73) PGAN CelebA  1K 0.99 0.99 6.51(1.85) — 19.05(3.14) PGAN CclebA 10K 0.98 0.98 5.05(1.63) — 16.75(1.87) CycleGAN Cityscapes  1K 0.99 0.99 55.85(3.67) — — CycleGAN Cityscapes 10K 0.99 0.99 49.66(5.01) — —

TABLE 7 Distinguishabilit (top), attributability (btm) before (Bfr) and after (Aft) robust training. DCGAN-M: DCGAN for MNIST, DCGAN-C: DCGAN for CelebA. GANs C Blurring Cropping Noise JPEG Combination — — Bfr Aft Bfr Aft Bfr Aft Bfr Aft Bfr Aft DCGAN-M  10 0.49 0.96 0.52 0.99 0.85 0.99 0.54 0.99 0.50 0.66 DCGAN-M 100 0.49 0.61 0.51 0.98 0.76 0.98 0.53 0.99 0.50 0.52 DCGAN-M  1K 0.49 0.50 0.51 0.81 0.69 0.91 0.53 0.97 0.50 0.51 DCGAN-C  10 0.49 0.99 0.49 0.99 0.95 0.98 0.51 0.99 0.50 0.85 DCGAN-C 100 0.50 0.96 0.49 0.49 0.92 0.93 0.50 0.99 0.49 0.61 DCGAN-C  1K 0.50 0.62 0.49 0.97 0.88 0.91 0.50 0.99 0.49 0.51 PGAN 100 0.50 0.98 0.51 0.99 0.97 0.99 0.96 0.99 0.50 0.76 PGAN  1K 0.50 0.89 0.49 0.95 0.94 0.95 0.88 0.99 0.50 0.60 PGAN 10K 0.50 0.61 0.50 0.76 0.89 0.90 0.76 0.98 0.50 0.51 CycleGAN 1K 0.49 0.92 0.49 0.87 0.98 0.99 0.55 0.99 0.49 0.67 CycleGAN 10K 0.49 0.70 0.50 0.66 0.94 0.96 0.52 0.98 0.50 0.51 DCGAN-M  10 0.49 0.96 0.49 0.97 0.85 0.98 0.53 0.99 0.49 0.65 DCGAN-M 100 0.50 0.54 0.49 0.97 0.74 0.96 0.52 0.94 0.49 0.52 DCGAN-M  1K 0.50 0.50 0.49 0.80 0.68 0.89 0.52 0.89 0.49 0.50 DCGAN-C  10 0.50 0.99 0.50 0.99 0.95 0.99 0.51 0.99 0.50 0.85 DCGAN-C 100 0.50 0.96 0.49 0.99 0.92 0.93 0.50 0.99 0.49 0.61 DCGAN-C  1K 0.49 0.61 0.50 0.98 0.87 0.89 0.50 0.99 0.50 0.51 PGAN 100 0.50 0.97 0.50 0.99 0.96 0.98 0.96 0.99 0.50 0.76 PGAN  1K 0.50 0.87 0.50 0.95 0.93 0.94 0.86 0.99 0.49 0.59 PGAN 10K 0.50 0.60 0.50 0.77 0.88 0.89 0.76 0.97 0.50 0.51 CycleGAN  1K 0.50 0.92 0.50 0.86 0.97 0.98 0.54 0.99 0.50 0.67 CycleGAN 10K 0.50 0.70 0.50 0.66 0.92 0.94 0.52 0.98 0.49 0.51

TABLE 8 ∥Δ.r∥ (top) and FID score (btm). Standard deviations in parenthesis. DCGAN-M: DCGAN for MNIST, DCGAN-C: DCGAN for CelebA. FID score not applicable to CycleGAN. ∥Δ.r∥ and FID score in Table 6 are baseline. Lower is better. GANs C Baseline Blurring Cropping Noise JPEG Combination DCGAN-M  10 5.20(0.31) 15.96(2.18) 9.17(0.65) 5.93(0.34) 6.48(0.94) 17.08(1.86) DCGAN-M 100 4.09(0.53) 12.95(4.47) 7.62(1.55) 4.57(0.78) 4.70(1.02) 12.70(3.37) DCGAN-M  1K 3.88(0.60) 7.17(2.10) 7.43(1.37) 4.22(0.77) 5.12(1.94) 7.56(1.41) DCGAN-C  10 4.19(0.18) 11.83(0.65) 9.30(0.31) 4.75(0.17) 6.01(0.29) 13.69(0.59) DCGAN-C 100 3.08(0.27) 10.00(1.61) 7.80(0.58) 3.20(0.45) 4.26(0.59) 11.65(1.48) DCGAN-C  1K 2.55(0.36) 7.68(1.53) 7.13(0.47) 2.65(0.24) 3.39(0.58) 9.23(1.22) PGAN 100 9.29(0.95) 18.49(2.04) 21.27(0.81) 10.20(0.81) 10.08(1.03) 24.82(2.33) PGAN  1K 6.52(1.85) 14.79(4.15) 18.88(1.96) 6.40(1.48) 7.09(1.62) 22.09(2.12) PGAN 10K 5.04(1.63) 10.19(2.87) 18.23(0.94) 5.13(1.14) 5.67(1.62) 17.26(1.39) CycleGAN  1K 55.85(3.67) 68.03(3.62) 80.03(3.59) 55.47(1.60) 57.42(2.00) 83.94(4.66) CycleGAN 10K 49.66(5.01) 58.64(3.70) 66.05(3.47) 53.14(0.44) 54.52(2.30) 66.24(5.29) DCGAN-M  10 5.68(0.23) 41.11(20.43) 21.58(2.44) 5.79(0.19) 6.50(1.70) 68.16(24.67) DCGAN-M 100 5.32(0.11) 23.83(14.29) 18.39(3.70) 5.41(0.18) 5.46(0.11) 36.05(16.20) DCGAN-M  1K 5.23(0.12) 10.85(4.28) 18.08(1.77) 5.37(0.14) 5.30(0.96) 21.86(4.16) DCGAN-C  10 52.09(2.20) 73.62(6.70) 98.86(9.51) 59.51(1.60) 60.35(2.57) 87.29(9.29) DCGAN-C 100 45.02(3.37) 73.12(11.03) 85.50(12.25) 47.60(2.57) 50.48(4.58) 78.11(12.95) DCGAN-C  1K 40.85(3.41) 55.63(7.97) 72.11(13.81) 40.87(3.03) 45.46(5.03) 57.13(7.20) PGAN 100 21.62(1.73) 28.15(3.43) 47.94(5.71) 25.43(2.19) 22.86(2.06) 45.16(7.87) PGAN  1K 19.05(3.14) 25.19(5.26) 43.48(12.24) 19.20(2.96) 19.05(2.82) 35.07(8.72) PGAN 10K 16.75(1.87) 18.96(2.65) 37.01(8.74) 16.94(1.89) 17.39(2.33) 26.63(4.44)

Referring to FIG. 11, a computer-implemented system (system) 1000 is shown including a processor 1002 configured with instructions 1004 for performing the attribution functionality described herein. In some embodiments, the processor 1002 is in operable communication with a data source 1006 that provides one or more datasets for generative models. As indicated, the processor 1002 is configured, via the instructions 1004, to provide a GAN 1008 and at least one key 1010 to each of a plurality of end user devices 1012; e.g., the end user devices 112 may download the GAN 1008 and a key 1010 via an application or app (e.g., app 1211 in FIG. 13). In some embodiments, the instructions 1004 may be implemented as code and/or machine-executable instructions executable by the processor 1002 that may represent one or more of a procedure, a function, a subprogram, a program, a routine, a subroutine, a module, an object, a software package, a class, or any combination of instructions, data structures, or program statements, and the like. In other words, the instructions 1004 described herein may be implemented by hardware, software, firmware, middleware, microcode, hardware description languages, or any combination thereof. When implemented in software, firmware, middleware or microcode, the program code or code segments to perform the necessary tasks (e.g., a computer-program product) may be stored in a computer-readable or machine-readable medium (e.g., the memory 1204 of FIG. 13), and the processor 1002 performs the tasks defined by the code. In some embodiments, the processor 1002 as configured, in combination with the data source 1006, collectively forms a registry and verification service 1014, and the processor 1002 as configured by the instructions 1004 can identify whether a query belongs to a particular GAN 1008. In general, as described, installation of the GANs 1008 to the end user devices 1012 as described modifies the GAN 1008 according to the keys 1010 so that each resulting model can be verified.

Referring to FIG. 12, an exemplary process 1100 is shown for exemplifying attribution concepts described herein. Referring to blocks 1102 and 1104, a processor (e.g., 1002) accesses a dataset and a base GAN model, and computes a plurality of keys. One or more keys may be used to further compute a key-dependent GAN model from the base GAN model. In blocks 1106 and 1108, the a plurality of such key-dependent models can be computed and each may be provided to users of end user devices so that each user trains a different, verifiable, key-dependent model.

As indicated in block 1110, each key-dependent GAN model can be verified or attributed by, e.g., leveraging a linear classifier that returns positive only for outputs of the respective model. In other words, each key-dependent model is parameterized by keys (distributed by a registry or otherwise). The keys may be computed from first-order sufficient conditions for decentralized attribution. The keys may further be orthogonal or opposite to each other and belong to a subspace dependent on the data distribution and the architecture of the generative model.

Exemplary Computing Device:

Referring to FIG. 13, a computing device 1200 is illustrated which may be configured, via one or more of an application 1211 or computer-executable instructions, to execute functionality described herein. More particularly, in some embodiments, aspects of the attribution methods and functions herein may be translated to software or machine-level code, which may be installed to and/or executed by the computing device 1200 such that the computing device 1200 is configured to execute functionality described herein. It is contemplated that the computing device 1200 may include any number of devices, such as personal computers, server computers, hand-held or laptop devices, tablet devices, multiprocessor systems, microprocessor-based systems, set top boxes, programmable consumer electronic devices, network PCs, minicomputers, mainframe computers, digital signal processors, state machines, logic circuitries, distributed computing environments, and the like.

The computing device 1200 may include various hardware components, such as a processor 1202, a main memory 1204 (e.g., a system memory), and a system bus 1201 that couples various components of the computing device 1200 to the processor 1202. The system bus 1201 may be any of several types of bus structures including a memory bus or memory controller, a peripheral bus, and a local bus using any of a variety of bus architectures. For example, such architectures may include Industry Standard Architecture (ISA) bus, Micro Channel Architecture (MCA) bus, Enhanced ISA (EISA) bus, Video Electronics Standards Association (VESA) local bus, and Peripheral Component Interconnect (PCI) bus also known as Mezzanine bus.

The computing device 1200 may further include a variety of memory devices and computer-readable media 1207 that includes removable/non-removable media and volatile/nonvolatile media and/or tangible media, but excludes transitory propagated signals. Computer-readable media 1207 may also include computer storage media and communication media. Computer storage media includes removable/non-removable media and volatile/nonvolatile media implemented in any method or technology for storage of information, such as computer-readable instructions, data structures, program modules or other data, such as RAM, ROM, EEPROM, flash memory or other memory technology, CD-ROM, digital versatile disks (DVD) or other optical disk storage, magnetic cassettes, magnetic tape, magnetic disk storage or other magnetic storage devices, or any other medium that may be used to store the desired information/data and which may be accessed by the computing device 1200. Communication media includes computer-readable instructions, data structures, program modules, or other data in a modulated data signal such as a carrier wave or other transport mechanism and includes any information delivery media. The term “modulated data signal” means a signal that has one or more of its characteristics set or changed in such a manner as to encode information in the signal. For example, communication media may include wired media such as a wired network or direct-wired connection and wireless media such as acoustic, RF, infrared, and/or other wireless media, or some combination thereof. Computer-readable media may be embodied as a computer program product, such as software stored on computer storage media.

The main memory 1204 includes computer storage media in the form of volatile/nonvolatile memory such as read only memory (ROM) and random access memory (RAM). A basic input/output system (BIOS), containing the basic routines that help to transfer information between elements within the computing device 1200 (e.g., during start-up) is typically stored in ROM. RAM typically contains data and/or program modules that are immediately accessible to and/or presently being operated on by processor 1202. Further, data storage 1206 in the form of Read-Only Memory (ROM) or otherwise may store an operating system, application programs, and other program modules and program data.

The data storage 1206 may also include other removable/non-removable, volatile/nonvolatile computer storage media. For example, the data storage 1206 may be: a hard disk drive that reads from or writes to non-removable, nonvolatile magnetic media; a magnetic disk drive that reads from or writes to a removable, nonvolatile magnetic disk; a solid state drive; and/or an optical disk drive that reads from or writes to a removable, nonvolatile optical disk such as a CD-ROM or other optical media. Other removable/non-removable, volatile/nonvolatile computer storage media may include magnetic tape cassettes, flash memory cards, digital versatile disks, digital video tape, solid state RAM, solid state ROM, and the like. The drives and their associated computer storage media provide storage of computer-readable instructions, data structures, program modules, and other data for the computing device 1200.

A user may enter commands and information through a user interface 1240 (displayed via a monitor 1260) by engaging input devices 1245 such as a tablet, electronic digitizer, a microphone, keyboard, and/or pointing device, commonly referred to as mouse, trackball or touch pad. Other input devices 1245 may include a joystick, game pad, satellite dish, scanner, or the like. Additionally, voice inputs, gesture inputs (e.g., via hands or fingers), or other natural user input methods may also be used with the appropriate input devices, such as a microphone, camera, tablet, touch pad, glove, or other sensor. These and other input devices 1245 are in operative connection to the processor 1202 and may be coupled to the system bus 1201, but may be connected by other interface and bus structures, such as a parallel port, game port or a universal serial bus (USB). The monitor 1260 or other type of display device may also be connected to the system bus 1201. The monitor 1260 may also be integrated with a touch-screen panel or the like.

The computing device 1200 may be implemented in a networked or cloud-computing environment using logical connections of a network interface 1203 to one or more remote devices, such as a remote computer. The remote computer may be a personal computer, a server, a router, a network PC, a peer device or other common network node, and typically includes many or all of the elements described above relative to the computing device 1200. The logical connection may include one or more local area networks (LAN) and one or more wide area networks (WAN), but may also include other networks. Such networking environments are commonplace in offices, enterprise-wide computer networks, intranets and the Internet.

When used in a networked or cloud-computing environment, the computing device 1200 may be connected to a public and/or private network through the network interface 1203. In such embodiments, a modem or other means for establishing communications over the network is connected to the system bus 1201 via the network interface 1203 or other appropriate mechanism. A wireless networking component including an interface and antenna may be coupled through a suitable device such as an access point or peer computer to a network. In a networked environment, program modules depicted relative to the computing device 1200, or portions thereof, may be stored in the remote memory storage device.

Certain embodiments are described herein as including one or more modules. Such modules are hardware-implemented, and thus include at least one tangible unit capable of performing certain operations and may be configured or arranged in a certain manner. For example, a hardware-implemented module may comprise dedicated circuitry that is permanently configured (e.g., as a special-purpose processor, such as a field-programmable gate array (FPGA) or an application-specific integrated circuit (ASIC)) to perform certain operations. A hardware-implemented module may also comprise programmable circuitry (e.g., as encompassed within a general-purpose processor or other programmable processor) that is temporarily configured by software or firmware to perform certain operations. In some example embodiments, one or more computer systems (e.g., a standalone system, a client and/or server computer system, or a peer-to-peer computer system) or one or more processors may be configured by software (e.g., an application or application portion) as a hardware-implemented module that operates to perform certain operations as described herein.

Accordingly, the term “hardware-implemented module” encompasses a tangible entity, be that an entity that is physically constructed, permanently configured (e.g., hardwired), or temporarily configured (e.g., programmed) to operate in a certain manner and/or to perform certain operations described herein. Considering embodiments in which hardware-implemented modules are temporarily configured (e.g., programmed), each of the hardware-implemented modules need not be configured or instantiated at any one instance in time. For example, where the hardware-implemented modules comprise a general-purpose processor configured using software, the general-purpose processor may be configured as respective different hardware-implemented modules at different times. Software may accordingly configure the processor 1202, for example, to constitute a particular hardware-implemented module at one instance of time and to constitute a different hardware-implemented module at a different instance of time.

Hardware-implemented modules may provide information to, and/or receive information from, other hardware-implemented modules. Accordingly, the described hardware-implemented modules may be regarded as being communicatively coupled. Where multiple of such hardware-implemented modules exist contemporaneously, communications may be achieved through signal transmission (e.g., over appropriate circuits and buses) that connect the hardware-implemented modules. In embodiments in which multiple hardware-implemented modules are configured or instantiated at different times, communications between such hardware-implemented modules may be achieved, for example, through the storage and retrieval of information in memory structures to which the multiple hardware-implemented modules have access. For example, one hardware-implemented module may perform an operation, and may store the output of that operation in a memory device to which it is communicatively coupled. A further hardware-implemented module may then, at a later time, access the memory device to retrieve and process the stored output. Hardware-implemented modules may also initiate communications with input or output devices.

Computing systems or devices referenced herein may include desktop computers, laptops, tablets e-readers, personal digital assistants, smartphones, gaming devices, servers, and the like. The computing devices may access computer-readable media that include computer-readable storage media and data transmission media. In some embodiments, the computer-readable storage media are tangible storage devices that do not include a transitory propagating signal. Examples include memory such as primary memory, cache memory, and secondary memory (e.g., DVD) and other storage devices. The computer-readable storage media may have instructions recorded on them or may be encoded with computer-executable instructions or logic that implements aspects of the functionality described herein. The data transmission media may be used for transmitting data via transitory, propagating signals or carrier waves (e.g., electromagnetism) via a wired or wireless connection.

It should be understood from the foregoing that, while particular embodiments have been illustrated and described, various modifications can be made thereto without departing from the spirit and scope of the invention as will be apparent to those skilled in the art. Such changes and modifications are within the scope and teachings of this invention as defined in the claims appended hereto. 

What is claimed is:
 1. A system for decentralized attribution of generative models, comprising: a processor configured with instructions to provide a registry and verification service that improves attribution of a generative adversarial network (GAN) relative to other versions of the GAN, such that the processor: accesses a dataset and a GAN associated with the dataset, computes a plurality of keys and a plurality of corresponding GANs such that at least one key is computed for each GAN of the plurality of GANs, each GAN of the plurality of GANs being a version of the GAN modified by the at least one key, wherein the plurality of keys are derived from first-order sufficient conditions for decentralized attribution, and verifies a GAN of the plurality of GANs based upon an output of a query associated with the GAN.
 2. The system of claim 1, wherein the plurality of keys are orthogonal keys that achieve distinguishability and attributability.
 3. The system of claim 1, wherein the plurality of keys belong to a subspace dependent upon data distribution and architecture of each GAN.
 4. The system of claim 1, wherein the processor is configured to provide access to each of the plurality of GANs to respective end user devices such that the respective end user devices train key-dependent models from the dataset.
 5. The system of claim 1, wherein each of the plurality of keys is held privately by a user associated with an end user device, the end user device being provided with a unique key-dependent GAN model.
 6. The system of claim 1, wherein each of the plurality of GANs is parameterized by the plurality of keys such that a linear classifier returns positive only for outputs of that model.
 7. A method for decentralized attribution of generative models, comprising: accessing, by a processor, information associated with a dataset, the dataset configured for GAN model generation for a plurality of end user devices; computing, by the processor one or more keys and a corresponding GAN model for each of the one or more keys to derive a plurality of GAN models, each of the plurality of GAN models being modified according to at least one key of the one or more keys, wherein each of the one or more keys is computed from first-order sufficient conditions for decentralized attribution; and verifying, by the processor, attribution of a model of the plurality of GAN models by implementing a linear classifier that returns positive only for outputs of the model.
 8. The method of claim 7, further comprising: providing access of the plurality of GAN models to respective end user devices such that each respective end user device is provided with a unique key-dependent model.
 9. The method of claim 8, further comprising installing an application to the respective end user device, the application including the GAN model and a key of the one or more keys, the installation modifying the GAN model according to the key to output a modified version of the GAN model verifiable by the key.
 10. A processing element, configured to: compute a sequence of keys by a registry, the keys configured for strict data compliance and orthogonality so as to accommodate tracing of machine-generated contents back to its source model, wherein the keys are orthogonal or opposite to each other and belong to a subspace dependent on data distribution and architecture of the generative model. 